Solve each proportion.

b30

39

=1. 5635

y5

=2.

412

p9

=3. 56m

2826

=4.

b = 10 y = 8

p = 3 m = 52

Warm Up

Pre-Algebra

7.6

Similar Figures

Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions in similar figures.

similar

Vocabulary

The heights of letters in newspapers and on billboards are measured using points and picas. There are 12 points in 1 pica and 6 picas in one inch.

A letter 36 inches tall on a billboard would be 216 picas, or 2592 points. The first letter in this paragraph is 12 points.

Points and Picas

Congruent figures have the same size and shape. Similar figures have the same shape, but not Similar figures have the same shape, but not necessarily the same sizenecessarily the same size. The A’s in the table are similar. The have the same shape, but they are not the same size.

The ratio formed by the corresponding sides is the scale factor.

Similar Figures

A picture 10 in. tall and 14 in. wide is to be scaled to 1.5 in. tall to be displayed on a Web page. How wide should the picture be on the Web page for the two pictures to be similar?To find the scale factor, divide the known measurement of the scaled picture by the corresponding measurement of the original picture.

0.15 0.151.510

=

Then multiply the width of the original picture by the scale factor.2.1 14 • 0.15 = 2.1

The picture should be 2.1 in. wide.

Example: Using Scale Factors to Find Missing Dimensions

A painting 40 in. tall and 56 in. wide is to be scaled to 10 in. tall to be displayed on a poster. How wide should the painting be on the poster for the two pictures to be similar?To find the scale factor, divide the known measurement of the scaled painting by the corresponding measurement of the original painting.0.25 0.2510

40=

Then multiply the width of the original painting by the scale factor.14 56 • 0.25 = 14

The painting should be 14 in. wide.

Try This

A T-shirt design includes an isosceles triangle with side lengths 4.5 in, 4.5 in., and 6 in. An advertisement shows an enlarged version of the triangle with two sides that are each 3 ft. long. What is the length of the third side of the triangle in the advertisement?

Set up a proportion.

6 in.x ft

4.5 in.3 ft

=

4.5 in. • x ft = 3 ft • 6 in. Find the cross products.

4.5 in. • x ft = 3 ft • 6 in. in. • ft is on both sides

Example: Using Equivalent Ratios to Find Missing Dimensions

4.5x = 3 • 6

4.5x = 18

x = = 4184.5

Cancel the units.

Multiply

Solve for x.

The third side of the triangle is 4 ft long.

Example Continued

Set up a proportion.

24 ftx in.

18 ft4 in.

=

18 ft • x in. = 24 ft • 4 in. Find the cross products.

18 ft • x in. = 24 ft • 4 in. in • ft is on both sides

A flag in the shape of an isosceles triangle with side lengths 18 ft, 18 ft, and 24 ft is hanging on a pole outside a campground. A camp t-shirt shows a smaller version of the triangle with two sides that are each 4 in. long. What is the length of the third side of the triangle on the t-shirt?

Try This

18x = 24 • 4

18x = 96

x = 5.39618

Cancel the units.

Multiply

Solve for x.

The third side of the triangle is about 5.3 in. long.

Try This Continued

The following are matching, or corresponding:A and X AB and XY

BC and YZ

AC and XZ

A

C BZ Y

X

C and Z

B and Y

Remember

Which rectangles are similar?

Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

Example: Identifying Similar Figures

50 48

The ratios are equal. Rectangle J is similar to rectangle K. The notation J ~ K shows similarity.

The ratios are not equal. Rectangle J is not similar to rectangle L.

20 = 20

length of rectangle Jlength of rectangle K

width of rectangle Jwidth of rectangle K

10 5

42

? =

length of rectangle Jlength of rectangle L

width of rectangle Jwidth of rectangle L

1012

45

?=

Compare the ratios of corresponding sides to see if they are equal.

Example Continued

Which rectangles are similar?

A8 ft

4 ft

B6 ft

3 ft

C5 ft

2 ft

Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

Try This

16 20

The ratios are equal. Rectangle A is similar to rectangle B. The notation A ~ B shows similarity.

The ratios are not equal. Rectangle A is not similar to rectangle C.

24 = 24

length of rectangle Alength of rectangle B

width of rectangle Awidth of rectangle B

8 6

43

? =

length of rectangle Alength of rectangle C

width of rectangle Awidth of rectangle C

8 5

42

?=

Compare the ratios of corresponding sides to see if they are equal.

Try This

Use the properties of similar figures to answer each question.

1. A rectangular house is 32 ft wide and 68 ft long. On a blueprint, the width is 8 in. Find the length on the blueprint.

2. Karen enlarged a 3 in. wide by 5 in. tall photo into a poster. If the poster is 2.25 ft wide, how tall is it?

3. Which rectangles are similar?

17 in.

3.75 ft

A and B are similar.

Lesson Quiz